Abstract
In this study we investigate the sharp radius of starlikeness of subclasses of Ma and Minda class for the ratio of analytic functions which are related to limaçon functions. This survey is connected also to the first-order differential subordinations. In this context, we get the condition on for which certain differential subordinations associated with limaçon functions imply Ma and Minda starlike functions. Simple corollaries are provided for certain examples of our results. Finally, we present several geometries related to our study.
1. Introduction and Preliminaries
Let be the class of normalized analytic functions having the seres form
Let denotes the class of all functions such that . The case is the usual class of Carathéodory functions. Let denotes the subclasses of consisting of functions which are univalent in . We say is subordinate to (written as or ) if there exists a Schwarz function such that for all . The class is one of the most vital categories of geometric function theory due to its wide applications in sciences and engineering. For example, univalent functions are extensively used in ODEs and PDEs and operators’ theory. Also, they are important in image processing techniques. Among the earlier subclasses of that had recieved tremendous attention are the classes and of convex and starlike functions, respectively.
In 1992, Ma and Minda [1] gave a unified characterization of the subclasses of , which consist of functions that map onto starlike domains. For this purpose, they considered analytic functions with in and normalized by and . Thus, the Ma and Minda class of starlike functions denoted by was defined by the subordination
In particular, for , the class reduces to the class of Janowski starlike functions [2]. The class is a well-known class of starlike functions of order . For , of starlike functions. More recently, many researchers have been investigating subclasses of having nice geometries in the right-half plane. In this direction, let be defined by
Then, for , the class reduces to , , , , and . The geometric properties of these classes have been demonstrated in [3–10] and the references therein.
In 2020, Masih and Kanas [11] explored another novel subclass of with . This class was denoted by and functions in it map onto a region bounded by limaçon. Saliu et al. [12] furthered the investigation of this class and obtained the bounds of the Hankel determinants, sharp radius, and differential implications associated with it. Also, Kanaga and Ravichandran [13] examined and found the smallest disc and the largest disc centered at such that
This concept was then applied to find the radius of limaçon starlikeness for . For more findings associated with , we refer to [14–16].
Let and be two subclasses of defined on . The radius of denoted by is the largest radius such that implies the function , defined by , for all . Radius result with ratio of analytic functions satisfying was studied by MacGregor [17], and the ones satisfying was examined by Ratti [18]. Presently, the radius of of various choices of for ratio of analytic functions has attracted the interest of researchers. To this end, Ali et al. [19] considered the functions whose ratio ,, and are each subordinate to or , for some analytic functions and . They obtained various radii of starlikeness for these classes. In clases where these are subordinate to or , Yadav et al. [20] also obtained various radii of starlikeness. Zhang et al. [21] found the radius of starlikeness connected with subclasses of for the ratio of analytic functions and satisfying
The theory of first-order differential subordinations arose from the work of Goluzin and Robertson in 1935 and 1947, respectively. Later, Miller and Mocanu [22, 23] developed and generalized this idea. Using this theory, many results related with the Ma and Minda class have emerged in different directions and perspectives in the literature. For more information in this direction, we refer to the recent work of Cho et al. [24] and Kumar and Gangania [25] with the references therein.
Let be the class of analytic functions satisfying the subordination:
Motivated with these aforecited works, we initiated the following classes of analytic functions:
Then, we investigate the sharp radius of starlikeness for various subclasses of Ma and Minda class. Moreover, the first-order differential subordination implications are also studied. Some special cases of our findings are given as a simple corollaries. Finally, we illustrate the geometries of some of our findings.
The following lemmas are required for our investigations.
Lemma 1. Let . Then, This bound is sharp for with .
Proof. From the property of subordination, we have , where is a Schwarz function. Therefore, a simple computation and Schwarz lemma ([26], p. 166) give Let with . Then, where . It is easy to see that is continous on . Then, by the elementary theorem of Real analysis, we have that where denotes the range of . Therefore,
Lemma 2 (see [23], Theorem 3.4h, p. 132). Let be univalent in and let and be analytic in a domain containing with , where . Set , and suppose that either (i) is convex or is starlike univalent in (ii)If is analytic in with and then , and is the best dominant in the sense that for all .
2. The Class
Define the function by
It is easy to see that and, thus, . Also, the class contains nonunivalent functions, since varnishes at . Hence, the radius of univalence for is for . It is observed that for in Corollary 4 (i), this radius coincides with the radius of starlikeness for the class.
Let ; then, there exists such that for . Therefore, a computation yields
Then, from Lemma 1, it follows that
3. The Class
Let the function be defined by
Then, the functions and satisfy for some . Thus, and . Since the class contains some nonunivalent functions. Hence, the radius of univalence for this class is .
Let . Then, there exists an analytic function such that for some . Since the disc implies , then . Let be defined such that
Then, and a computation gives
It is known from Lemma 2 of [27] that for , we have
Using this fact for the case and Lemma 1 in (27), we arrive at
It also follows from (29) that provided . For , we have at . This shows that the radius is sharp. Hence, the radius of starlikeness for the class is the same as the radius of univalence for the class.
4. The Class
Let . Then, . It is obvious that contains nonunivalent functions, since varnishes at for . Hence, the radius of univalence for is for . We notice that this radius coincides with the radius of starlikess of the class for the choice of in Corollary 4 (iii).
Let be defined such that for ,
Then, an obvious calculation yields
In view of Lemma 1, we arrive at
5. Radius of Starlikeness
In this section, we obtain the radii of starlikeness of the classes , , and for different Ma and Minda starlike classes of functions.
Silverman ([28], pp. 50-51) showed that holds if and only if . Using this fact, we obtain the radii of Janowski starlikeness for and .
Theorem 3. The following sharp results hold for Janowski starlike class (i)(ii)
Proof. (i)Let . Then, satisfies (20). To prove our result, it suffices to show that Then, from (36), it follows that where we have that For in (16), we have At , (ii)Let . Then, satisfies (35). We need to show that Then, from (36), it follows that which gives For in Section 4 at , we have which implies
Corollary 4. The following sharp results hold for (i)(ii)(iii)
Proof. The proof of (i) and (iii) are direct from Theorem 3 for
and . For (ii), provided (30) satisfies
That is, . Then, and . Therefore, there exists such that , where we obtain given in the theorem. For sharpness, consider the function in (21). Then, for , we have that
The class is the class of parabolic starlike functions introduced by Rønning [8]. This class consists of functions satisfying the inequality
For , Shanmugam and Ravichandran [29] proved the following inclusion relation for the class .
Theorem 5. The following sharp results hold for (i)(ii)(iii)
Proof. (i)For , relation (50) implies that disc (20) lies in provided or if . The function shows that the radius of parabolic starlike functions is sharp. For this function, we have At , a calculation shows that (ii)Since the center of disc (29) is , then it stays inside if or equivalently . For at , a computation gives (iii)Since the center of disc (35) is , then inclusion (50) holds provided which implies . For the function at , we have
The class of the exponential starlike functions satisfies the inequality
This class was initiated by Mendiratta et al. [6] in 2015. They also proved that for ,
Theorem 6. The following sharp results hold for (i)(ii)(iii)
Proof. (i)For , relation (59) implies that disc (20) lies inside whenever Equivalently, if . The function shows that this radius is best possible since and at , we have (ii)Since , inclusion (59) implies that disc (29) is in provided which at the same time means At for , we arrive at (iii)For , disc (35) lies completely in if or . Thus, the of the function is . The sharpness is seen from the function at , i.e.,
In 2016, Sharma et al. [9] introduced the class of functions that map the open unit disc onto a Cardiod domain. A function if it satisfies the subordination:
The following inclusion relation was also established in [9]: provided , where
Theorem 7. The following sharp results hold for the class (i)(ii)(iii)
Proof. (i)Since the center of (20) is , then by relation (69), we have that belongs to provided or . This radius cannot be improved since the function assumes the extremum, i.e., at , (ii)From (69), it follows that disc (29) stays inside whenever This gives . To prove that this radius is sharp, we consider the function such that at , we arrive at (iii)Let . Then, we see that if which gives . For the function , we see that which shows that the radius is sharp
In 2019, Cho et al. [4] considered and investigated the class . They also proved the following: if , where is the image of under the function .
Theorem 8. The following sharp results hold for (i)(ii)(iii)
Proof. (i)It is easy to see from (77) that disc (20) stays inside the region provided which implies that . To show that this radius is best possible, we consider the function so that At , we arrive at (ii)Since , then disc (29) satisfies relation (77) provided This gives For the function , the result is sharp. Indeed, at , we have (iii)Proceeding as in the above cases and using inclusion (77), we find that if which holds for . The sharpness of the radius is assumed by the function
Kumar and Arora [5] investigated the subclass of Ma and Minda class of functions that map onto a petal domain. They also proved the inclusion relation where is the image of under the function . In view of the procedure of Theorem 8, (20), (29), and (35), we have the following theorem.
Theorem 9. The following sharp results hold for (i)(ii)(iii)
In 2021, Bano and Mohsan [3] introduced the subclass of analytic functions characterized by the subordination
They also proved that where is the image domain of under the mapping .
Theorem 10. The following sharp results hold for (i)(ii)(iii)
Proof. (i)Since disc (20) has a center , then relation (86) implies that this disc lies inside whenever or . The sharpness of this radius is achieved by the function at (ii)For , disc (29) is inside the domain provided which means that The function shows that the radius cannot be improved since at , we have (iii)It is observed that disc (35) is in if which equivalently implies . For the function , we see that at ,
6. Differential Subordination Implications
In this section, we present some first-order differential subordinations associated with .
Theorem 11. Let be analytic in with , and suppose Then, we have the following subordination results(i), for and with and (ii)(iii)(iv) for (v)(vi)(vii)These bounds are sharp.
Proof. Consider the analytic function such that is a solution of the differential equation Let and . Then, the function is defined by such that This means that is a starlike univalent function in . It is easy to see that the function satisfies Therefore, by Lemma 2, we have that Each subordinations of Theorem 11 is equivalent to for each superordinate function in the theorem, which holds if . Then, This yields the necessary condition for which . Looking at the geometry of each of the functions , it is observed that this condition is also sufficient. (i)Let . Then, We notice that Thus, (ii). Then, the inequalities and reduce to and . We note that . Thus, (iii)Let . Then, from (100), we have We notice that . Therefore, the subordination holds if (iv)Let . Then, the inequalities and give and , where and . A calculation shows that . Therefore, the subordination holds if (v)Let . Then, the inequalities and give and , where and . A calculation shows that . Therefore, the subordination holds provided (vi)Let . Then, following the same line of proof of , we show that the subordination holds provided (vii)Let . Then, the inequalities and yield and . A calculation shows that . Therefore, the subordination holds provided
As a direct consequence of Theorem 11, we have the following results.
Corollary 12. Let and suppose it satisfies the following subordination then (i) provided with where (ii) provided (iii) provided (iv) provided (v) provided (vi) provided (vii) provided
Theorem 13. Let be analytic in with , and suppose Then, the following subordination results are true: (i), for and with (ii) for (iii) for (iv) for (v) for (vi) for (vii) for These bounds on cannot be improved.
Proof. Let the analytic function be defined by Then, is a solution of the differential equation Let and . Then, the function is defined by . As it was demonstrated in Theorem 11, we note that is starlike in and that satisfies Therefore, by Lemma 2, we have that Each subordinations of Theorem 11 is equivalent to for each superordinate function in the theorem. From here, we obtain the required results by following the same line of proof as in Theorem 11.
The following results are direct consequences of Theorem 13
Corollary 14. Let and suppose it satisfies the following subordination then (i) provided , where for , (ii) provided (iii) provided (iv) provided (v) provided (vi) provided (vii) provided
Theorem 15. Let be analytic in with , and suppose Then, the following subordination results hold(i), for and with and (ii) for (iii) for (iv) for (v) for (vi) for (vii) for These bounds on are sharp.
Proof. Consider the analytic function such that Then, is a solution of the differential equation Let and . Then, the function is defined by . A computation shows that is starlike in . Also, we have that satisfies Therefore, by Lemma 2, we have that Hence, we obtain the desired results by following the same line of proof of Theorem 11
From Theorem 15, we have the following consequences.
Corollary 16. Let and suppose it satisfies the following subordination then (i) provided with where (ii) provided (iii) provided (iv) provided (v) provided (vi) provided (vii) provided
The following geometries illustrate the sharpness of the Corollaries 12, 14 and 16 from (ii) to (vii) as we can see in Figure 1.
7. Conclusion
We introduced the classes , , and and then obtained the sharp radii for which normalized analytic functions in these classes belonged to Ma and Minda starlike class. Moreover, applying the well-known theory of differential subordination developed by Miller and Mocanu, we found the restriction on for which the differential subordination implies certain Ma and Minda subclasses. We gave some special cases of our findings. Lastly, geometries of parts of our investigations were also illustrated.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.